Ellipsoid‐Based Interval‐Type Uncertainty Model Updating Based on Riemannian Manifold and Gaussian Process Model

Modern engineering systems require advanced uncertainty‐aware model updating methods that address parameter correlations beyond conventional interval analysis. This paper proposes a novel framework integrating Riemannian manifold theory with Gaussian Process Regression (GPR) for systems governed by Symmetric Positive‐Definite (SPD) matrix constraints. Our methodology features three key innovations: (1) A semi‐definite programming‐optimized Minimum Volume Ellipsoid model that explicitly quantifies parameter interdependencies while ensuring computational efficiency; (2) A manifold‐embedded GPR surrogate model employing Log‐Euclidean kernels to intrinsically preserve SPD constraints during uncertainty updating; (3) A Riemannian gradient optimization scheme that enables efficient parameter updates via logarithmic matrix mapping. Validated through mechanical and aerospace case studies, the framework achieves a Log‐Euclidean distance of 3.12 × 10 −4 in uncertainty updating (compared to a baseline of 48.48) and provides a tenfold computational acceleration over Bayesian alternatives. Robustness tests demonstrate stable performance under 5% noise perturbation, with the Log‐Euclidean distance increased only marginally to 1.41 × 10 −2 . By unifying differential geometry with machine learning, our approach eliminates heuristic projections required in conventional methods while advancing uncertainty quantification through structure‐preserving manifold operations. This study bridges geometric consistency, computational efficiency, and physical consistency in uncertainty‐aware model updating.